Vector Spaces Space of vectors, closed under addition and scalar multiplication

Image Averaging as Vector addition

Scaler product, dot product, norm

Norm of Images

Orthogonal Images, Distance,Basis

Roberts Basis: 2x2 Orthogonal

Cauchy Schwartz Inequality  U+V  ≤  U  +  V 

Schwartz Inequality

Quotient: Angle Between two images

Fourier Analysis

Fourier Transform Pair Given image I(x,y), its fourier transform is

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Complex Arithmetic

Fourier Traansform of an Image is a complex matrix Let F =[F(u,v)] F = Φ MM I(x,y) Φ NN I(x,y)= Φ* MM F Φ* MM Where Φ JJ (k,l)= [Φ JJ (k,l) ] and Φ JJ (k,l) = (1/J) exp(2Πjkl/J) for k,l= 0,…,J-1

Fourier Transform

Properties Convolution Given the FT pair of an image f(x,y) F(u,v) and mask pair h(x,y) H(u,v) f(x,y)* h(x,y) F(u,v). H(u,v) and f(x,y) h(x,y) F(u,v)* H(u,v)

Properties of Fourier Transform

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Design of H(u,v) İdeal Low Pass filter H(u,v) = 1 if |u,v |< r 0 o.w. Ideal High pass filter H(u,v) = 1 if |u,v |> r 0 o.w Ideal Band pass filter H(u,v) = 1 if r1<|u,v |< r2 0 o.w

İmage Enhancement Spatial Smoothing Low Pass Filtering

Ideal Low pass filter

Ideal Low Pass Filter

Output of the Ideal Low Pass Filter

Gaussian Low Pass Filyer

Gaussian Low Pass Filter

High Pass Filter: Ideal and Gaussian

Ideal High Pass

Fourier Transform-High Pas Filtering

Frequency Spectrum of Damaged Circuit

Gaussian Low Pass and High Pass

Output of Gaussian High Pass

Gaussian Filters: Space and Frequency Domain

Spatial Laplacian Masks and its Fourier Transform

Laplacian Filter

Laplacian Filtering